9780203009512.ch10

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139 Element and Global Stiffness and Mass Matrices 10.1 APPLICATION OF THE PRINCIPLE OF VIRTUAL WORK Elements of variational calculus were discussed in Chapter 3, and the Principle of Virtual Work was introduced in Chapter 5. Under static conditions, the principle is repeated here as (10.1) As before, δ represents the variational operator. We assume for our purposes that the displacement, the strain, and the stress satisfy representations of the form (10.2) in which E and S are written as one-dimensional arrays in accordance with traditional ﬁnite-element notation. For use in the Principle of Virtual Work, we need D , which introduces the factor 2 into the entries corresponding to shear. We suppose that the boundary is decomposed into four segments: S = S I + S II + S III + S IV . On S I , u is prescribed, in which event u vanishes. On S II , the traction τ is prescribed as τ 0 . On S III , there is an elastic foundation described by τ = τ 0 A ( x ) u , in which A ( x ) is a known matrix function of x . On S IV , there are inertial boundary conditions, by virtue of which τ = τ 0 . The term on the right now becomes (10.3) 10 δδ ρ τ ESdV u u dV u dS ij ij i i i i ∫∫∫ += ˙˙ . ux ) x T (( ) ( ) ( )( ) ,t t , , == = ϕ Φ γ β Φ γ ES E T xD δτ ud S d S dS t dS t ii ∫∫ = ++ γ Φ ϕτ γ Φ ϕ ϕΦγ γ Φ ϕ TT 0 T T x xA x xB x () . SS S S S II III IV III IV © 2003 by CRC CRC Press LLC

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140 Finite Element Analysis: Thermomechanics of Solids The term on the left in Equation 10.1 becomes (10.4) in which K is called the stiffness matrix and M is called the mass matrix. Canceling the arbitrary variation and bringing terms with unknowns to the left side furnishes the equation as follows: (10.5) Clearly, elastic supports on S III furnish a boundary contribution to the stiffness matrix, while mass on the boundary segment S IV furnishes a contribution to the mass matrix. Sample Problem 1: One element rod Consider a rod with modulus E, mass density ρ , area A , and length L . It is built in at x = 0. At x = L , there is a concentrated mass m to which is attached a spring of stiffness k , as illustrated in Figure 10.1. The stiffness and mass matrices, from the domain, reduce to the scalar values K EA / L , M AL / 3, M S m , K S k . The governing equation is . Sample Problem 2: Beam element Consider a one-element model of a cantilevered beam to which a solid disk is welded at x = L . Attached at L is a linear spring and a torsional spring, the latter having the property that the moment developed is proportional to the slope of the beam. FIGURE 10.1 Rod with inertial and compliant boundary conditions. δ δρ ESdV ud V ij ij ii ∫∫ == δγ γ Φ β βΦ γ ρ Φϕϕ Φ TT T T KK x D x MM x x () , ( ) ˙˙ , ( ) ( ) , t dV ut dV ( ) ( ) f ++ + = SS tt γ γ fx Kx A x Mx B x T 0 = = = + Φϕτ Φϕ ϕΦ . dS dS dS S S S S II III III IV ( ) EA L AL km f + = γγ 3 E,A,L, ρ P m k © 2003 by CRC CRC Press LLC
Element and Global Stiffness and Mass Matrices 141 The shear force V 0 and the moment M 0 act at L . The interpolation model, incorpo- rating the constraints w (0, t ) = w (0, t ) = 0 a priori , is (10.6)

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9780203009512.ch10 - 0749_Frame_C10 Page 139 Wednesday 6:04...

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